pull-in retarding in nonlinear nanoelectromechanical resonators under … · 2017. 2. 3. ·...
TRANSCRIPT
Pull-In Retarding in Nonlinear Nanoelectromechanical
Resonators Under Superharmonic Excitation
Najib Kacem, Sebastien Baguet, Sebastien Hentz, Regis Dufour
To cite this version:
Najib Kacem, Sebastien Baguet, Sebastien Hentz, Regis Dufour. Pull-In Retarding in Nonlin-ear Nanoelectromechanical Resonators Under Superharmonic Excitation. Journal of Compu-tational and Nonlinear Dynamics, American Society of Mechanical Engineers (ASME), 2012, 7(2), pp.021011. <10.1115/1.4005435>. <hal-00726024>
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Pull-In Retarding in Nonlinear Nanoelectromechanical Resonators UnderSuperharmonic Excitation
Najib Kacem∗
Universite de LyonCNRS INSA-Lyon
LaMCoS UMR5259
F-69621 Villeurbanne FranceEmail: [email protected]
Sebastien BaguetUniversite de LyonCNRS INSA-Lyon
LaMCoS UMR5259
F-69621 Villeurbanne FranceEmail: [email protected]
Sebastien HentzCEA/LETI - MINATEC,
17 rue des MartyrsF-38054 Grenoble, France
Email: [email protected]
Regis DufourUniversite de Lyon
CNRS INSA-Lyon
LaMCoS UMR5259F-69621 Villeurbanne France
Email: [email protected]
In order to compensate for the loss of performance when
scaling resonant sensors down to NEMS, a complete analyt-
ical model including all main sources of non linearities is
presented as a predictive tool for the dynamic behavior of
clamped-clamped nanoresonators electrostatically actuated.
The nonlinear dynamics of such NEMS under superharmonic
resonance of order half their fundamental natural frequen-
cies is investigated. It is shown that the critical amplitude
has the same dependence on the quality factor Q and the
thickness h as the case of the primary resonance. Finally,
a way to retard the pull-in by decreasing the AC voltage is
proposed in order to enhance the performance of NEMS res-
onators.
∗Address all correspondence to this author.
1 Introduction
The small size of NEMS resonators combined with their
physical attributes make them quite attractive and suitable for
a wide range of technological applications such as ultrasen-
sitive force and mass sensing, narrow band filtering, and time
keeping. However, at this size regime, nonlinearities occur
sooner which reduces the dynamic range of such devices.
It is a challenge to achieve large-amplitude motion of
NEMS resonators without deteriorating their frequency sta-
bility [1]. The relative frequency noise spectral density [2]
of a NEMS resonator is given by:
S f =
(
1
2Q
)2Sx
P0(1)
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where Sx is the displacement spectral density and P0 is the
displacement carrier power, ie the RMS drive amplitude of
the resonator 12A2. Remarkably, driving the resonator at
large oscillation amplitude leads to better signal to noise ra-
tio (SNR) and, thus, simplifies the design of the electronic
feedback loop. However, doing so in the nonlinear regime
reduces the sensor performances since its frequency stabil-
ity becomes dependent on its oscillation amplitude. More-
over, even when NEMS resonators are used as oscillators in
closed-loop, a large part of noise mixing [3, 4] due to non-
linearities drastically reduces their dynamic range and alters
their detection limit.
A nonlinear model for a clamped-clamped microbeam
was introduced in a previous work [5] and the ability to sup-
press the hysteresis of the nonlinear frequency response un-
der primary resonance was demonstrated by third order non-
linearity cancellation. Thus the performances of resonant
sensors might be enhanced by driving the resonator beyond
its critical amplitude [6–8].
Nevertheless, the operating domain of the third order
nonlinearity cancellation is limited by the occurrence of the
mixed behavior [7–10] which appears proportionally sooner
with respect to the resonator size and closer to the critical
amplitude. Moreover, at this oscillation level and in the case
of electrostatic forces actuation, pull-in (the upper bound
limit) is easily reachable which makes the electrical charac-
terization of NEMS resonators quite difficult and limits the
possible domain of resolution enhancement.
In order to avoid such complications, we investigate the
dynamics of NEMS resonators under superharmonic reso-
nance. Jin and Wang [11] showed that driving a microbeam
of a resonant microsensor by a superharmonic excitation of
order one-half increases the signal-to-crosstalk ratio as com-
pared to driving it at primary resonance.
The dynamic behavior of MEMS resonators under sec-
ondary resonances has been investigated by many authors.
Turner et al [12] studied the response of a comb-drive device
to a parametric excitation that offers interesting behavior, and
a possibility for novel applications such as parametric am-
plification [13–15] and noise squeezing [13]. Younis and
Nayfeh [16] and Abdel-Rahman and Nayfeh [17] used the
method of multiple scales to study the response of an electro-
statically deflected microbeam based resonator to a primary-
resonance excitation, a superharmonic-resonance excitation
of order two, and a subharmonic-resonance excitation of or-
der one-half. Since they are based on perturbation methods,
these models yield accurate results only for small AC ampli-
tudes and hence small motions.
Younis et al [18] and Nayfeh and Younis [19] studied the
global dynamics of MEMS resonators under superharmonic
excitation and showed that the dynamic pull-in phenomenon
can occur for a superharmonic excitation at an electric load
much lower than that predicted by a static analysis.
In this paper, the dynamics of a nanoelectromechanical
resonator, actuated around half its fundamental natural fre-
quency, is simulated. A complete analytical model incorpo-
rating all main sources of nonlinearities is used based on the
Galerkin discretisation coupled with multiple scales pertur-
bation technique.
The dependences of the resonator critical amplitude on
the physical parameters are analyzed in order to compare it
with the primary resonance case. The results are shown with
respect to the quality factor and actuation voltages. Finally,
the dynamic behavior of the resonator up to the pull-in insta-
bility [19, 20] is tracked and a solution to enhance the res-
onator performances is provided.
2 Model
Existing models [16–20] for the nonlinear dynamics of
MEMS resonators with relatively high capacitive variations
concern designs with only one electrode for both actuation
and sensing. NEMS resonators though, have low capacitive
variations, and it is almost necessary to use a two port mea-
surement, i.e. to separate detection and actuation electrodes
in order to enhance the signal to background ratio. Moreover,
the use of different gaps (gd < ga) enables the maximization
of the detection signal.
The resonator considered here is presented in Fig. 1.
It consists of a clamped-clamped microbeam actuated by an
electric load v(t) =−Vdc +Vac cos(Ωt), where Vdc is the DC
polarization voltage, Vac is the amplitude of the applied AC
voltage, and Ω is the forcing frequency.
Fig. 1. Schema of an electrically actuated microbeam
2.1 Equation of motion
The transverse deflection of the microbeam w(x, t) is
governed by the nonlinear Euler-Bernoulli equation, which
is the commonly used approximate equation of motion for a
thin beam [21]
EI∂4w(x, t)
∂x4+ρbh
∂2w(x, t)
∂t2+ c
∂w(x, t)
∂t
−[
N +Ebh
2l
∫ l
0
[
∂w(x, t)
∂x
]2
dx
]
∂2w(x, t)
∂x2
=1
2ε0
bCn1
[
Vac cos(Ωt)−Vdc
]2
(ga − w(x, t))2H1(x)
−1
2ε0
bCn2V 2dc
(gd + w(x, t))2H2(x) (2)
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H1(x) = H
(
x− l + la
2
)
−H
(
x− l − la
2
)
(3)
H2(x) = H
(
x− l + ld
2
)
−H
(
x− l − ld
2
)
(4)
where x is the position along the microbeam length, c is
the linear viscous damping per unit length, E and I are the
Young’s modulus and moment of inertia of the cross sec-
tion. N is the applied tensile axial force due to the residual
stress on the silicon, t is time, ρ is the material density, h
is the microbeam thickness, ga and gd are respectively the
actuation and the sensing capacitor gap thickness, ε0 is the
dielectric constant of the gap medium. The last term in Eq.
(2) represents an approximation of the electric force assum-
ing a resonator design with 2 stationary electrodes : the first
one is devoted to actuation while the second one is used for
sensing. The fringing field effect [22] was taken into account
using the coefficients Cni. Since the electrodes do not act on
the whole length of the beam, the electrostatic force distri-
butions are modeled by means of Heaviside functions H(x).The microbeam is subject to the following boundary condi-
tions
w(0, t) = w(l, t) =∂w
∂x(0, t) =
∂w
∂x(l, t) = 0 (5)
2.2 Normalization
For convenience and equations simplicity, the following
nondimensional variables are introduced
w =w
gd
, x =x
l, t =
t
τ(6)
where τ =2l2
h
√
3ρ
E.
Substituting Eq. (6) into Eqns. (2) and (5) yields
∂4w
∂x4+
∂2w
∂t2+ c
∂w
∂t+α2Cn2
V 2dc
(1+w)2H2(x)
= α2Cn1[Vac cos(Ωt)−Vdc]
2
(Rg −w)2H1(x)
+
[
N +α1
∫ 1
0
[
∂w
∂x
]2
dx
]
∂2w
∂x2(7)
w(0, t) = w(1, t) =∂w
∂x(0, t) =
∂w
∂x(1, t) = 0 (8)
The parameters appearing in Eq. (7) are
H2(x) = H
(
x− l + la
2l
)
−H
(
x− l− la
2l
)
(9)
H2(x) = H
(
x− l + ld
2l
)
−H
(
x− l − ld
2l
)
(10)
c =cl4
EIτ, N =
Nl2
EI, α1 = 6
[ga
h
]2
(11)
Rg =ga
gd
, α2 = 6ε0l4
Eh3g3a
, Ω = Ωτ (12)
2.3 Solving
The beam total displacement w(x, t) can be written as
a sum of a static dc displacement ws(x) and a time-varying
ac displacement wd(x, t). However, in the present case, the
static deflection can be considered as negligible because the
measured quality factors Q are in the range of 2.103 − 104
and Vdc ≤ 40Vac. Indeed, the ratio between the static and
the dynamic deflections is
ws(x)
wd(x, t)≈ Vdc
2Q.Vac≤ 1% (13)
A reduced-order model is generated by modal decompo-
sition, transforming Eq. (7) into a finite-degree-of-freedom
system consisting in ordinary differential equations in time.
The undamped linear mode shapes of the straight microbeam
are used as basis functions in the Galerkin procedure, and the
deflection is approximated by
w(x, t) =n
∑k=1
ak(t)φk(x) (14)
where n is the number of retained modes (size of the modal
basis), ak(t) is the kth generalized coordinate and φk(x) is the
kth linear undamped mode shape of the straight microbeam.
The linear undamped mode shapes φk(x) are governed by:
d4φk(x)
dx4= λ2
kφk(x) (15)
φk(0) = φk(1) = φ′k(0) = φ′k(1) (16)
The analytical form of the eigenmodes is given by
φk(x) = Ak
[
coshλk − cosλk
sinλk − sinhλk
]
[sinλkx− sinhλkx]
+cosλkx− coshλkx
(17)
with λk solutions of the transcendental equation
1− cosλk coshλk = 0 (18)
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These functions are a modal basis for the scalar product
〈u,v〉=∫ 1
0u(x)v(x)dx (19)
and the coefficients Ak are chosen to normalize the eigen-
modes such that 〈φi,φ j〉 = δi j, where δi j is the Kronecker
symbol.
Let Eq. (7) be multiplied by φk(x) [(1+w)(Rg −w)]2
in order to include the complete contribution of the nonlin-
ear electrostatic forces in the resonator dynamics without ap-
proximation. This method, which has been used by Nayfeh
et al. [19, 20] for a one-port resonator, has some disadvan-
tages like the non orthogonality of the operator w4 ∂4w∂x4 with
respect to the undamped linear mode shapes of the resonator,
the increase of the nonlinearity level in the normalized equa-
tion of motion (7) as well as the incorporation of new nonlin-
ear terms such as the Van der Pol damping. Nevertheless, the
resulting equation contains less parametric terms than if the
nonlinear electrostatic forces were expanded in Taylor series
and the solution of nonlinear problem is valid for large dis-
placements of the beam up to the sensing gap.
Substituting Eqns. (14) and (15) into the resulting equa-
tion, and then integrating the outcome from x = 0 to x = 1
yields a system of coupled ordinary differential equations in
time.
For several nonlinear mechanical systems such as shells
[23], many modes are needed in order to build the frequency
response without losing any physical information concerning
the nonlinear coupling between the modes. Nevertheless, in
our case (simple beam resonators), the first mode can be con-
sidered as the dominant mode of the system and the higher
modes can be neglected. This has been demonstrated nu-
merically using time integration [7] as well as shooting and
harmonic balance method coupled with an asymptotic nu-
merical continuation technique [24]. Thus, only one mode is
considered (n = 1).
Moreover, since Vac << Vdc and consequently V 2ac <<
Vac.Vdc , the effects of the fast harmonics (proportional to
V 2ac) on the slow excitations (proportional to Vac.Vdc ) are neg-
ligible [25]. Thus, the linear and nonlinear terms related to
the second harmonic has not been considered and the follow-
ing equation is obtained:
a1 + ca1 +ωn2a1 + µ1a1a1 + µ2a1
2a1 + µ3a13a1
+µ4a14a1 + cµ1a1a1 + cµ2a1
2a1 + cµ3a13a1
+cµ4a14a1 +χ2a1
2 +χ3a13 +χ4a1
4 +χ5a15
+χ6a16 +χ7a1
7 +ν+ ζ0 cos(Ωt)
+ζ1a1 cos(Ωt)+ ζ2a12 cos(Ωt) = 0 (20)
Some canonical nonlinear terms can be identified in Eq. (20),
such as the Duffing non linearity (χ3a13), the nonlinear Van
der Pol damping (cµ2a12a1) as well as the parametric exci-
tations (ζ1a1 cos(Ωt)+ ζ2a12 cos(Ωt)) which are due to the
fact that the beam is actuated by two electrodes. Indeed, for
one-port resonators, the parametric terms are eliminated by
the pre-multiplication of the equation of motion by the de-
nominator of the electrostatic force [24].
Other terms correspond to high-level nonlinearities and
nonlinear parametric excitations. All these terms come from
the coupling between the mechanical and the electrostatic
nonlinearities as well as the nonlinear coupling between both
electrostatic forces due to the pre-multiplication of Eq. (7) by
the common denominator of both electrostatic forces in ac-
tuation and sensing. The appendix shows the expressions of
all the integration parameters appearing in Eq. (20), which
can be easily computed with any computational software.
To analyse this equation of motion, it proves convenient
to invoke perturbation techniques which work well with the
assumptions of ”small” excitation and damping (Q > 10),
typically valid in MEMS resonators. In this paper, the
method of multiple scales [26] is used to attack Eq. (20)
in order to determine a uniformly valid approximate solu-
tion. To this end, we seek a first-order uniform solution in
the form
a1(t,ε) = a10(T0,T1)+ εa11(T0,T1)+ · · · (21)
where ε is the small nondimensional bookkeeping parameter,
T0 = t and T1 = εt. Since the non linear response to a super-
harmonic resonance excitation of order two is analyzed, the
nearness of Ω to ωn2
is expressed by introducing the detuning
parameter σ according to
2Ω = ωn + εσ (22)
Substituting Eq. (21) into Eq. (20) and equating coefficients
of like powers of ε yields
Order ε0
cos
(
σT1 +T0ωn
2
)
ζ0 +ω2na10 + a10
(2,0) = 0 (23)
Order ε1
a210χ2 + a3
10χ3 + a410χ4 + a5
10χ5 + a610χ6 + a7
10χ7
+a10cµ1a10(1,0)+ a2
10cµ2a10(1,0)+ a3
10cµ3a10(1,0)
+cos(σT1 +T0ωn)ζ3 + cos(σT1 +T0ωn)a10ζ4
+a410cµ4a10
(1,0)+ a10µ1a10(2,0)+ a2
10µ2a10(2,0)
+a11ω2n + cosa2
10ζ5 (σT1 +T0ωn)+ ca10(1,0)
+2a10(1,1)+ a3
10µ3a10(2,0)+ a4
10µ4a10(2,0)
+a11(2,0)+ cos
(
σT1 +T0ωn
2
)
a10ζ1
+cos
(
σT1 +T0ωn
2
)
a210ζ2 = 0 (24)
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where a( j,k)i =
∂k
∂kT k1
(
∂ j
∂ jTj
0
)
.
The general solution of Eq. (23) can be written as
a01 = Acos(ωnT0 +Φ)− 4ζ0
3ω2n
cos
(
ωnT0
2+σT1
)
(25)
Equation (25) is then substituted in Eqns. (24) and the
trigonometric functions are expanded. The elimination of
the secular terms yields two first order non-linear ordinary-
differential equations which describe the amplitude and
phase modulation of the response and permit a stability anal-
ysis
A = f1(ε,A,β)+O(ε2) (26)
β = f2(ε,A,β)+O(ε2) (27)
where β = 2σT1 −Φ. The steady-state motions occur when
A = β = 0, which corresponds to the fixed points of Eqns.
(26) and (27). Thus, the frequency-response equation can be
written in its parametric form with respect to the phase β as
a set of two equations
A = K1(β) (28)
Ω = K2(β) (29)
This analytic expression makes the model suitable for
MEMS and NEMS designers as a fast and efficient tool for
resonant sensor performances optimisation. The expressions
of Eqns. (28) and (29) are quite large. For the sake of con-
ciseness, they are not detailed here
3 Results
All the numerical computations leading to Eqns. (28)
and (29) were carried out with the following set of param-
eters: l = 50µm, b = 0.5µm, la = 40µm, ga = 500nm,
ld = 48µm, gd = 300nm, while h, Vac and Vdc were used
for parametric analysis.
As shown in Fig. 2, the analytical model enables the
capture of all the nonlinear phenomena in the resonator dy-
namics and describes the competition between the hardening
(presented by the mechanical nonlinearity) and the softening
(presented by the electrostatic nonlinearity) behaviors. Re-
markably, the mixed behavior [9] was not observed. Indeed,
the superharmonic excitation filters out the effect of the fifth
order nonlinear terms and the compensation of the nonlin-
earities by hysteresis suppression [7, 8] is possible at large
amplitudes. The upper bound limit of such operation is obvi-
ously the pull-in detailed below, but first the bistability limit
(critical amplitude) of the resonator under superharmonic ex-
citation is investigated by comparison to the primary reso-
nance case.
Fig. 2. Competition between hardening and softening behaviors for
different values of the ratioh
gd(Wmax is the normalized displacement
at the middle of the beam)
3.1 Critical amplitude
As shown in Fig. 3, the critical amplitude is the oscilla-
tion amplitude Ac above which bistability occurs. Thus, Ac is
the transition amplitude from the linear to the nonlinear be-
havior. For the primary resonance of a clamped-clamped mi-
Fig. 3. Forced frequency responses of the typical resonator de-
scribed in Fig. 1 for h = 0.5µm and Q = 5000. σ is the detuning
parameter and Wmax is the displacement of the beam normalized by
the gap gd at its middle pointl2
. B1 and B2 are the two bifurcation
points of a typical hardening behavior.
crobeam, Ac is defined as the oscillation amplitude for which
the equationdΩ
dβ= 0 has a unique solution βc =
2π
3[27]. It
was also shown analytically and experimentally [7, 28] that
the critical amplitude for micromechanical clamped-clamped
beam resonators under primary resonance is only determined
by the beam thickness h in the direction of vibration and the
quality factor Q. The question is whether or not the critical
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amplitude of a NEMS resonator is invariant with respect to
the excitation.
In order to explain how to deduce the mechanical crit-
ical amplitude from Eqns. (28) and (29), we assume the
simplified case of neglected nonlinear electrostatic effects
(h
gd
<< 1). Using Eq. (29), the derivative of the frequency
Ω with respect to the phase β is directly deduced as a func-
tion of several variables:
dΩ
dβ= Ω′(χi,ξ0,ωn,β) (30)
One can search the condition for which the equation
Ω′(χi,ξ0,ωn,β) = 0 has a unique solution β. After some
algebraic transformations, a relation between χi, ωn and ξ0
has been identified and the critical amplitude of the electro-
static force ξ0c is expressed as a function of the design and
nonlinear parameters of the resonator. Explicitly, when the
electrostatic forcing is ξ0 = ξ0c, Ω′ is null only at the critical
phase βc =2π3
.
The critical amplitude is the value of Eq. (28) at the
phase β = π2
(for a critical behavior) multiplied by 1.588
which corresponds to the amplification coefficient of the first
mode at the middle of the clamped-clamped beam x = 12.
Replacing ξ0 by ξ0c in the resulting equation and multiply-
ing by the gap gd to obtain the dimensional value and using
the expressions of the needed parameters from the appendix,
yields:
Ac = 1.685h√Q
(31)
It can be noted that this analytic expression for the critical
amplitude is the same as in the case of primary resonance [7].
In order to validate this simple expression, some computa-
tions are carried out using Eqns. (28) and (29). The cor-
responding results are presented hereafter and the process
leading to the dependency of Ac on h and Q is explained
graphically. The AC voltage and the beam thickness are re-
spectively fixed at Vac = 0.6V and h = 1µm. Then, for three
values of DC voltage, dΩdβ is calculated at the phase βc =
2π3
and plotted with respect to the quality factor Q as shown in
Fig. 4a. The critical quality factor Qc corresponds to the
intersection between the curvedΩ( 2π
3 )
dβ [Q] and the Q-axis.
The critical amplitude is then the maximum amplitude
of the frequency response curves plotted in 4b for each of
the three values of the critical quality factor Q1, Q2 and
Q3. At this point the dependency between the critical am-
plitude and the quality factor can be plotted, as seen in
Fig. 4c. This permits to conclude that the critical ampli-
tude Ac decreases when the quality factor increases. More-
over, the identification of the coefficients Coe f1 and Coe f2
such that Log(Ac) =Coe f1−Coe f2Log(Q) leads to Coe f1 =Log(1.685 h
gd) = 0.448 and Coe f2 = − 1
2, which is in very
good agreement with Eq. (31).
Fig. 4. (a): Critical quality factor determination for different DC volt-
age (h = 500nm and Vac = 0.6V ). (b): Dependency of the critical
amplitude on the quality factor (Here Ac is the peak of Wmax). (c):
Logarithmic plot of the critical amplitude with respect to the quality
factor.
Figure 5a shows the variation of the critical thickness
hc for different DC polarizations Vdc. The fixed values of
the AC voltage and the quality factor Q are Vac = 0.6V and
Q = 4900. The critical thickness hc is determined as the in-
tersection with the horizontal axis in the same way as Qc.
These values are used to plot the frequency responses of Fig.
5b and then Figure 5c is obtained, which clearly indicates the
linear dependence of the critical amplitude Ac on the thick-
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ness of the resonator. Again, these numerical results are in
very good agreement with Eq. (31).
Fig. 5. (a): Critical thickness determination for different DC voltage
(Q = 4900 and Vac = 0.6V ). (b): Dependency of the critical ampli-
tude on the thickness (Here Ac is the peak of Wmax). (c): Plot of the
critical amplitude with respect to the beam thickness.
3.2 Pull-in retarding
In order to make the detection technique easier for
NEMS resonant sensors, it is convenient to drive the res-
onator at very large oscillations, i.e. beyond the critical am-
plitude Ac, which means in the nonlinear regime. However,
at this vibration level, instabilities such as pull-in are likely to
occur, which might cause the ruin of the sensor. Therefore, it
is crucial for NEMS designers to predict the bistability limit
(critical amplitude) as well as the upper bound amplitude not
to be exceeded in order to avoid pull-in. In the following
we not only propose a simple analytical expression for the
pull-in amplitude, but also discuss a way to retard it.
3.2.1 Pull-in amplitude
The NEMS resonator must operate in the safe work-
ing range and away from instabilities. One of the main
drawbacks with electrostatically actuated NEMS structures
is the so-called pull-in instability, i.e., the collapse of the mi-
crobeam on the electrode, which has been investigated with
both static [29] and dynamic [20, 30] applied voltages.
The pull-in amplitude is the oscillation amplitude above
which the resonator position becomes unstable and collapses.
Several researchers predicted the presence of dynamic pull-
in in various beam-based systems at voltages lower than the
static pull-in voltage [19, 20, 30–32]. Fargas-Marques et
al [33] used an energy-based analytical method for the dy-
namic pull-in prediction in electrostatic MEMS and provided
experimental data. Ashhab et al [34] and Basso et al [35]
have also investigated the problem of escape from potential
well for atomic force microscopes (AFM), with the attractive
force being the van der Waals forces.
Unlike these energy-based approaches, we investigate
the pull-in phenomenon via the dynamic stability of a fixed
point of the corresponding Poincare map which is obviously
a logical choice, given the developed reduced-order model
and the resulting amplitude and phase modulation equations.
More specifically, the initiation of the pull-in domain under
superharmonic resonance has been located at an amplitude
reaching an infinite slope at the phase βp = π2
which corre-
sponds to a Floquet multiplier approaching unity [7, 19, 20].
For convenience, the lower bound of this domain is called
”pull-in amplitude” since at this oscillation level which is
beyond the critical amplitude, the beam can collapse.
Figure 6 a) shows several predicted frequency-response
curves under superharmonic resonance. For the chosen de-
sign, the resonator operates in the nonlinear regime. As a
result, the response amplitude are much greater than the nor-
malized critical amplitude given by Acgd
= 0.1025 which is the
limit of the linear regime. For Vdc = 15V , the response curve
exhibits a classical nonlinear hardening behavior with a max-
imum normalized amplitude Wmax = 0.26. For Vdc = 20V ,
the response curve is highly nonlinear and the dynamic pull-
in occurs at Wmax ≃ 0.35 with a phase βp =π2
in Eqns. (28)-
(29). As shown on Figure 6 b), the response curve also
reaches an infinite slope at the pull-in amplitude. The res-
onator becomes highly unstable inside a narrow frequency
zone surrounding the pull in amplitude. This frequency band
corresponds to an escape zone where the kinetic energy of
the nonlinear resonator exceeds the barrier of its potential
well [38]. Inside this narrow zone, the nonlinear vibrations
of the beam follow a pull-in attractor causing the damage of
the NEMS sensor. This escape phenomenon is schematically
represented by a bold point. It is worth noting that the present
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reduced-order model is capable of predicting the pull-in phe-
nomenon with only one mode retained and a semi-analytical
solving, in contrast with the purely numerical method used
in [19].
Fig. 6. (a): Predicted frequency curves for different DC voltage
(Q = 3000, h = 1µm and Vac = 0.4V ). (b): Zoom on the escape
band and the infinite slope at the phase β = Pi2
for Vdc = 20V .
3.2.2 Pull-in control
Methods have been employed so far to control pull-in
phenomenon such the simple addition of a series capaci-
tance [36] as well as the ”voltage control algorithm”. The
latter has been applied by Chu and Pister [37]. It is based on
controlling the voltage according to a feedback loop that ex-
actly prevents the pull-in while taking into account the effect
of nonlinearity for getting more exact solutions to this insta-
bility. Unlike these techniques, we propose a simple electro-
static mechanism based on the increase of the ratio between
the applied DC and AC voltages.
Ideally, it is desirable to drive the resonator at large os-
cillations up to the gap gd in order to get the best signal to
noise ratio as possible. However, this is hardly compatible
with low dimensions devices and high quality factors accord-
ing to Eq. (31). As a result, the critical amplitude is often
very low and the operating range between the noise floor and
the critical amplitude is very narrow, i.e. the device is prone
to pull-in even at low displacements. As a consequence, res-
onators are generally designed to vibrate with amplitudes be-
low a small fraction (less than a tenth) of the gap to guarantee
the efficiency and no damage of the device. For further im-
provements of the sensors, MEMS and NEMS designers are
interested in maximizing the detected signals in order to en-
able the electric characterization of electrostatically actuated
nanoresonators with conventional measurement equipments.
The control of the pull-in amplitude is thus of prime interest
and the question is how to retard it.
In order to understand which parameters have an in-
fluence on the pull-in amplitude under superharmonic reso-
nance, the DC voltage is increased successively from 12V to
25V while lowering the AC voltage from 0.5V to 0.3V . Re-
markably, the unavoidable pull-in escape frequency band is
reached at larger oscillation amplitudes for low Vac and high
Vdc as shown in Fig. 7. The pull-in amplitude is increased
from 0.25 up to 0.45, which represents a 80% enhancement.
This pull-in retarding mechanism with respect to the res-
onator displacement is purely electrostatic and it is caused
by the amplification of the nonlinear electrostatic stiffness
which balances the mechanical hardening nonlinearities and
modifies the bifurcation topology of the expected dynamic
response. Indeed, increasing the ratioVdcVac
while keeping a
constant Vdc.Vac accelerates the spring softening effects and
consequently, it enhances the stable oscillations of the res-
onator below the pull-in amplitude.
Fig. 7. Predicted frequency curves (up to pull-in) for different AC
and DC polarizations (Q = 3000 and h = 1µm).
Figure 8 shows the variation of the resonant amplitude
up to pull-in with respect to AC voltage Vac. For NEMS res-
onators under superharmonic resonance, it is possible to shift
up the pull-in amplitude by applying a low AC voltage. It is
important to underline that this ability to control the pull-in
amplitude with only one physical parameter (the driving AC
voltage) is not possible under primary resonance [7]. How-
ever, in order to compensate for the loss of performance, the
8
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resonator must be actuated with higher DC polarizations.
Fig. 8. Dependency of the pull-in amplitude on the AC voltage.
4 Conclusion
A global approach to model and simulate the dynam-
ics of NEMS resonators is presented. Particularly, superhar-
monic resonance excitation of order-two is investigated and
results of analytical simulations are presented. They show
that the superharmonic excitation filters out the mixed be-
havior [9] and the third order nonlinearity cancellation [5, 7]
is only limited by the pull-in. Moreover, compared to the
primary resonance, it is demonstrated that the critical ampli-
tude of the resonator does not change and keeps the same
dependence on the quality factor Q and the thickness h. Fur-
thermore, the existence and the mechanism of the dynamic
pull-in phenomenon under superharmonic excitation are an-
alytically analyzed and close-form expressions for the pull-in
amplitude as well as the pull-in voltage could be provided us-
ing the model. In practice, this analytical model constitutes
a fast and efficient tool for NEMS designers for the evalu-
ation of the appropriate DC voltage and the amplitude and
frequency of the AC load in order to shift up or retard pull-
in while operating around the hysteresis suppression domain.
Thus the resonator performances can be enhanced since driv-
ing it linearly at large amplitude could prevent most of noise
mixing [3, 4].
Acknowledgements
The authors gratefully acknowledge financial support
from the CEA LETI and I@L Carnot institutes (NEMS
Project) and from the MNTEurop Project.
Appendix
ω2n =−2V 2
dc
Cn2α2
Rg
∫ l+ld2l
l−ld2l
φ12 dx
−Cn1α2
2V 2dc+V 2
ac
R2g
∫ l+la2l
l−la2l
φ12 dx
+λ21 + 12.3026N (32)
µ1 = 2.65876(1+Rg)
Rg
(33)
µ2 = 1.85191+ 4Rg+R2
g
R2g
(34)
µ3 = 5.302191+Rg
R2g
(35)
µ4 =−7.727211+Rg
R2g
(36)
χ2 = 1.32938
(
2λ21
(1+ 2Rg)
Rg
+Cn1α2V 2
ac
Rg2
)
+Cn2α2
(
V 2dc − 2VdcVs+Vs2
)
Rg2
∫ l+ld2l
l−ld2l
φ13 dx
−V 2dcCn1α2
Rg2
∫ l+la2l
l−la2l
φ13 dx
+38.2811N(1+ 2Rg)
Rg
(37)
χ3 = 28.2132N1+ 4Rg+R2
g
R2g
+ 151.354α1
+1.8519λ21
1+ 4Rg+R2g
R2g
(38)
χ4 = 470.958α11+Rg
Rg
+ 5.30219λ21
+83.1444N1+Rg
R2g
(39)
χ5 = 3.8636λ2
1
R2g
+ 61.6548N
R2g
+347.096α21
(
1+ 4Rg+R2g
)
(40)
χ6 = 1022.89α11+Rg
R2g
(41)
χ7 = 758.515α1
R2g
(42)
ν =−1
2
(
Cn1α2
(
V 2ac + 2V2
dc
)
∫ l+la2l
l−la2l
φ1 dx
)
+Cn2R2gα2
(
V 2dc +Vs2
)
∫ l+ld2l
l−ld2l
φ1 dx (43)
ζ0 = 4VacVdcCn1α2
∫ l+la2l
l−la2l
φ1 dx (44)
9
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ζ1 =−4VacVdcCn1α2
∫ l+la2l
l−la2l
φ21 dx (45)
ζ2 = 2VacVdcCn1α2
∫ l+la2l
l−la2l
φ31 dx (46)
References
[1] Feng, X. L., 2007. “Phase noise and frequency stabil-
ity of very-high frequency silicon nanowire nanome-
chanical resonators”. In 14th International Conference
on Solid-State Sensors, Actuators and Microsystems,
pp. 327–30.
[2] Robins, W. P., 1984. Phase Noise in Signal Sources.
Institution of Engineering and Technology.
[3] Roessig, T. A., Howe, R. T., and Pisano, A. P., 1997.
“Nonlinear mixing in surface-micromachined tuning
fork oscillators”. In Frequency Control Symposium,
1997., Proceedings of the 1997 IEEE International,
pp. 778–782.
[4] Kaajakari, V., Koskinen, J. K., and Mattila, T., 2005.
“Phase noise in capacitively coupled micromechani-
cal oscillators”. Ultrasonics, Ferroelectrics and Fre-
quency Control, IEEE Transactions on, 52(12), Dec.,
pp. 2322–2331.
[5] Kacem, N., Hentz, S., Fontaine, H., Nguyen, V.,
Robert, P., Legrand, B., and Buchaillot, L., 2008.
“From mems to nems: Modelling and characterization
of the non linear dynamics of resonators, a way to en-
hance the dynamic range”. In Int. Conf. Nanotech,
(Boston : Massachusetts, U.S.A).
[6] Postma, H. W. C., Kozinsky, I., Husain, A., and
Roukes, M. L., 2005. “Dynamic range of nanotube-
and nanowire-based electromechanical systems”. Ap-
plied Physics Letters, 86(22), p. 223105.
[7] Kacem, N., Hentz, S., Pinto, D., Reig, B., and Nguyen,
V., 2009. “Nonlinear dynamics of nanomechanical
beam resonators: improving the performance of nems-
based sensors”. Nanotechnology, 20(27), p. 275501.
[8] Kacem, N., Arcamone, J., Perez-Murano, F., and
Hentz, S., 2010. “Dynamic range enhancement of non-
linear nanomechanical resonant cantilevers for highly
sensitive nems gas/mass sensor applications”. Jour-
nal of Micromechanics and Microengineering, 20(4),
p. 045023.
[9] Kacem, N., and Hentz, S., 2009. “Bifurcation topol-
ogy tuning of a mixed behavior in nonlinear microme-
chanical resonators”. Applied Physics Letters, 95(18),
p. 183104.
[10] Kacem, N., Baguet, S., Hentz, S., and Dufour, R.,
2010. “Nonlinear phenomena in nanomechanical res-
onators: mechanical behaviors and physical limita-
tions”. Mecanique & Industries, 11(6), pp. 521–529.
[11] Jin, Z., and Wang, Y., 1998. “Electrostatic resonator
with second superharmonic resonance”. Sensors and
Actuators A: Physical, 64(3), pp. 273 – 279.
[12] Turner, K. L., Miller, S. A., Hartwell, P. G., MacDon-
ald, N. C., Strogatz, H. S., and Adams, S. G., 1998.
“Five parametric resonances in a microelectromechan-
ical system”. Nature, 396, pp. 149–152.
[13] Rugar, D., and Grutter, P., 1991. “Mechanical paramet-
ric amplification and thermomechanical noise squeez-
ing”. Phys. Rev. Lett., 67(6), Aug, pp. 699–702.
[14] Carr, D. W., Evoy, S., Sekaric, L., Craighead, H. G.,
and Parpia, J. M., 2000. “Parametric amplification in
a torsional microresonator”. Applied Physics Letters,
77(10), pp. 1545–1547.
[15] Carr, D. W., Evoy, S., Sekaric, L., Craighead, H. G.,
and Parpia, J. M., 1999. “Measurement of mechanical
resonance and losses in nanometer scale silicon wires”.
Applied Physics Letters, 75(7), pp. 920–922.
[16] Younis, M., and Nayfeh, A. H., 2003. “A study of the
nonlinear response of a resonant microbeam to an elec-
tric actuation”. Nonlinear Dynamics, 31(1), pp. 91–
117.
[17] Abdel-Rahman, E. M., and Nayfeh, A. H., 2003. “Sec-
ondary resonances of electrically actuated resonant mi-
crosensors”. Journal of Micromechanics and Micro-
engineering, 13(3), pp. 491–501.
[18] Younis, M. I., Abdel-Rahman, E. M., and Nayfeh,
A. H., 2004. “Global dynamics of mems resonators
under superharmonic excitation”. In the International
Conference on MEMS, NANO, and Smart Systems,
Banff, Canada (ICMENS), p. 694699.
[19] Nayfeh, A. H., and Younis, M. I., 2005. “Dynamics
of mems resonators under superharmonic and subhar-
monic excitations”. Journal of Micromechanics and
Microengineering, 15(10), pp. 1840–1847.
[20] Nayfeh, A. H., Younis, M. I., and Abdel-Rahman,
E. M., 2007. “Dynamic pull-in phenomenon in mems
resonators”. Nonlinear Dynamics, 48, pp. 153–163.
[21] Landau, L. D., and Lifshitz, E. M., 1986. Theory of
Elasticity. Butterworth-Heinemann.
[22] Nishiyama, H., and Nakamura, M., 1990. “Capacitance
of a strip capacitor”. Components, Hybrids, and Man-
ufacturing Technology, IEEE Transactions on, 13(2),
Jun, pp. 417–423.
[23] Touze, C., and Thomas, O., 2006. “Non-linear be-
haviour of free-edge shallow spherical shells: Effect
of the geometry”. International Journal of Non-Linear
Mechanics, 41(5), pp. 678–692.
[24] Kacem, N., Baguet, S., Hentz, S., and Dufour, R., 2011.
“Computational and quasi-analytical models for non-
linear vibrations of resonant mems and nems sensors”.
International Journal of Non-Linear Mechanics, 46(3),
pp. 532–542.
[25] Belhaq, M., and Fahsi, A., 2008. “2:1 and 1:1
frequency-locking in fast excited van der polmath-
ieuduffing oscillator”. Nonlinear Dynamics, 53,
pp. 139–152.
[26] Nayfeh, A. H., 1981. Introduction to Perturbation
Techniques. Wiley.
[27] Yurke, B., Greywall, D. S., Pargellis, A. N., and Busch,
P. A., 1995. “Theory of amplifier-noise evasion in an
oscillator employing a nonlinear resonator”. Phys. Rev.
A, 51(5), May, pp. 4211–4229.
10
Prod
uit p
ar H
AL -
22 S
ep 2
014
[28] Shao, L. C., Palaniapan, M., and Tan, W. W., 2008.
“The nonlinearity cancellation phenomenon in mi-
cromechanical resonators”. Journal of Micromechanics
and Microengineering, 18(6), p. 065014.
[29] Younis, M. I., Abdel-Rahman, E. M., and Nayfeh, A.,
2003. “A reduced-order model for electrically actu-
ated microbeam-based mems”. Microelectromechani-
cal Systems, Journal of, 12(5), oct., pp. 672–680.
[30] Krylov, S., and Maimon, R., 2004. “Pull-in dynamics
of an elastic beam actuated by continuously distributed
electrostatic force”. Journal of Vibration and Acoustics,
126(3), pp. 332–342.
[31] De, S. K., and Aluru, N. R., 2004. “Full-lagrangian
schemes for dynamic analysis of electrostatic mems”.
Microelectromechanical Systems, Journal of, 13(5),
oct., pp. 737–758.
[32] Elata, D., and Bamberger, H., 2006. “On the dy-
namic pull-in of electrostatic actuators with multiple
degrees of freedom and multiple voltage sources”. Mi-
croelectromechanical Systems, Journal of, 15(1), feb.,
pp. 131–140.
[33] Fargas-Marques, A., Casals-Terre, J., and Shkel, A. M.,
2007. “Resonant pull-in condition in parallel-plate
electrostatic actuators”. Microelectromechanical Sys-
tems, Journal of, 16(5), oct., pp. 1044–1053.
[34] Ashhab, M., Salapaka, M. V., Dahleh, M., , and Mezic,
I., November 1999. “Melnikov-based dynamical analy-
sis of microcantilevers in scanning probe microscopy”.
Nonlinear Dynamics, 20, pp. 197–220(24).
[35] Basso, M., Giarre, L., Dahleh, M., and Mezic, I., 2000.
“Complex dynamics in a harmonically excited lennard-
jones oscillator: Microcantilever-sample interaction in
scanning probe microscopes”. Journal of Dynamic Sys-
tems, Measurement, and Control, 122(1), pp. 240–245.
[36] Seeger, J. I., and Crary, S. B., 1997. “Stabilization of
electrostatically actuated mechanical devices”. In Solid
State Sensors and Actuators, 1997, Vol. 2, pp. 1133 –
1136 vol.2.
[37] Chu, P. B., and Pister, S. J., 1994. “Analysis of closed-
loop control of parallel-plate electrostatic microgrip-
pers”. In Robotics and Automation, 1994. Proceed-
ings., 1994 IEEE International Conference on, pp. 820
–825 vol.1.
[38] Gottwald, J. A., Virgin, L. N., and Dowell, E. H., 1995.
“Routes to escape from an energy well”. Journal of
Sound and Vibration, 187(1), pp. 133 – 144.
List of Figures
1 Schema of an electrically actuated microbeam 2
2 Competition between hardening and soften-
ing behaviors for different values of the ratioh
gd(Wmax is the normalized displacement at
the middle of the beam) . . . . . . . . . . . . 5
3 Forced frequency responses of the typical
resonator described in Fig. 1 for h = 0.5µm
and Q = 5000. σ is the detuning parame-
ter and Wmax is the displacement of the beam
normalized by the gap gd at its middle pointl2. B1 and B2 are the two bifurcation points
of a typical hardening behavior. . . . . . . . . 5
4 (a): Critical quality factor determination for
different DC voltage (h = 500nm and Vac =0.6V ). (b): Dependency of the critical am-
plitude on the quality factor (Here Ac is the
peak of Wmax). (c): Logarithmic plot of the
critical amplitude with respect to the quality
factor. . . . . . . . . . . . . . . . . . . . . . 6
5 (a): Critical thickness determination for dif-
ferent DC voltage (Q = 4900 and Vac =0.6V ). (b): Dependency of the critical am-
plitude on the thickness (Here Ac is the peak
of Wmax). (c): Plot of the critical amplitude
with respect to the beam thickness. . . . . . . 7
6 (a): Predicted frequency curves for different
DC voltage (Q = 3000, h = 1µm and Vac =0.4V ). (b): Zoom on the escape band and the
infinite slope at the phase β= Pi2
for Vdc = 20V . 8
7 Predicted frequency curves (up to pull-in) for
different AC and DC polarizations (Q= 3000
and h = 1µm). . . . . . . . . . . . . . . . . . 8
8 Dependency of the pull-in amplitude on the
AC voltage. . . . . . . . . . . . . . . . . . . 9
11
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